Workshop 7
Direct and Inverse Variation
Insights Into Algebra 1
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Workshop 7
Overview
Description
This workshop presents two activities through which teachers can give students experience in studying natural
events by using simulations. Students simulate oil spills on land and water, and develop mathematical models of
these phenomena. Through these models, they investigate direct and inverse variations.
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Part I: Peggy Lynn and her students simulate oil spills on land. The ratio of the number of drops of oil to the
resulting surface area of the spill is used to introduce direct variation.
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Part II: Peggy Lynn and her students simulate oil spills in water. Students investigate the relationship
between the volume of cylinders and the area of the base. The concept of inverse variation is developed in
this context.
Featured Textbook
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“Oil: Black Gold,” in SIMMS Integrated Mathematics: A Modeling Approach Using Technology; Level 1,Volume 2;
pp. 157-172. Simon & Schuster Custom Publishing, 1996.
Featured Educator
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Peggy Lynn, West Yellowstone High School; West Yellowstone, Montana
Featured Commentator
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Beatrice Moore-Harris, Associate Mathematics Manager, Project Grad Houston; Houston, Texas
Learning Objectives
In these activities, you will learn how to help students:
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Develop and graph direct and inverse proportions.
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Develop mathematical models of real-world events.
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Use mathematical models to make predictions about data sets.
Workshop 7
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Insights Into Algebra 1
Workshop Session (On-Site)
Part I: Direct Variation
Part II: Inverse Variation
Getting Ready (15 minutes)
Getting Ready (15 minutes)
Write down everything you know about direct variation. Identify situations that are modeled by direct
variation functions and share this information with
your group.
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List several examples of problem situations that
could be modeled by direct variation functions
and inverse variation functions.
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Divide into small groups, share your examples,
and collectively produce more examples.
Watching the Video (30 minutes)
Watching the Video (30 minutes)
Watch Part I: Direct Variation.
Watch Part II: Inverse Variation.
Going Further (15 minutes)
Select two or three of the questions listed below for
discussion during the session. You may want to discuss
the others on Channel-Talk or reflect on them in your
online journal.
Going Further (15 minutes)
Select two or three of the questions listed below for
discussion. You may want to discuss the others on
Channel-Talk or reflect on them in your online journal.
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What strategies from the video might you incorporate in your classroom? Discuss how you
might incorporate them.
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Discuss whether or not it is worthwhile to have
“messy numbers” when using real data in the
teaching of mathematics.
•
What should a teacher be looking and listening
for while students are working in groups? Discuss the order in which Peggy selected students
to present their work.
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Discuss the questioning techniques Peggy used
in this workshop. How do they enhance student
learning?
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How did Peggy capitalize on the student group
that used a linear function that wasn’t a direct
variation as a model? Discuss how she modified
the lesson as a result of their work.
•
Discuss how the teacher introduces vocabulary
words and formulas at the end of the lesson
rather than the beginning. Why does she do
this? What are the advantages of each method?
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In a direct variation, multiplying the x-coordinate by a scale factor increases the y-coordinate
by the same scale factor. Sometimes students
assume that this is true for all linear functions.
How did Peggy help students deal with this misconception?
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What are some possible next steps to this
lesson? How might you move beyond the
lesson in the video with your students?
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Design another experiment for students that
would be modeled by an inverse variation
function.
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What are some possible next steps to this
lesson? How might you move beyond the
lesson in the video?
Insights Into Algebra 1
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Workshop 7
Between Sessions (On Your Own)
Homework Assignment
Read Peggy Lynn’s thoughts and reflections in the Video Teacher Reflection section and discuss them on ChannelTalk. The reflections consist of Peggy’s responses to a set of questions generated by mathematics educators who
had viewed her video lessons. She wrote these responses after viewing the video herself.
Ongoing Activities
You may want to carry on these activities throughout the course of the workshop.
Keep a Journal
Read the Teaching Strategies for Workshop 7 and answer the journal prompts. Include thoughts, questions, and
discoveries from the workshop itself and learning experiences that take place in your own classroom. You are
encouraged to use the online journaling tool at www.learner.org/channel/workshops/algebra.
Web Site: www.learner.org/channel/workshops/algebra
Look at the Resources section for more ideas on questioning, lesson planning, and ways to teach the concepts of
direct variation and inverse variation.
Share Ideas on [email protected]
Share your thoughts about how you can improve your questioning techniques and lesson planning.
Workshop 7
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Insights Into Algebra 1
Video Teacher Reflections
Peggy Lynn
Below are Peggy Lynn’s responses to
some of the comments and questions raised by other mathematics
educators after they viewed the
workshop video:
What are your thoughts after watching the video?
Observing my class as an outsider looking in was
very educational for me. My first response was “I love
these activities!” It reaffirmed my dedication to using
context to teach mathematics—in other words, it
works! And the materials for these activities are easy to
find, easy to set up, and easy to clean up. It was fun to
watch how comfortable my students were with
expressing their ideas and using hand tools (graphing
scatterplots), as well as technology, to explore their
results. I also feel using their own data helps them to
realize that not every scatterplot is perfect, and that
not every slope is an integer.
There would be more guided practice and independent practice (class examples and homework) in a
normal class period than appeared on the video. I
would also like to emphasize the use of a “warm-up”
before every class to review the previous lesson and
prerequisite skills needed for the lesson that day.
Discuss your approach to questioning in each lesson.
I think it is critical for students to explain their
thinking—for correct and incorrect responses. My job
is to guide them through the process as they construct
the mathematical concepts desired. Asking how and
why, not just what, during discussions is helpful to get
students to communicate their thoughts. When a student answers a question incorrectly, I may suggest the
question that their response would have answered
correctly instead of just replying,“Wrong.”It takes time
to get students to be comfortable with making educated guesses—to be risk takers. There is a lot of
power in figuring it out for yourself.
Are there places where you wish you could have
changed a question or a follow-up question?
During the summary of both lessons, instead of just
writing the ideas on the board, I wish I had allowed the
students to contribute the key concepts.
Insights Into Algebra 1
How well does this lesson follow the lesson plan that
you wrote for it?
I have used these lessons several times, so there are
few surprises at this point. I do modify the chart for
gathering data from what the textbook suggests so it
lends itself better to the discussion I wish to have
happen.
Where did things go differently than you expected?
a. The question about whether or not to include
(0,0) when finding the mean ratio of
area/volume (drops) from the particular group
that asked was unexpected. But it gave me the
opportunity to discuss (briefly) division by zero.
b. Including a y-intercept other than zero in their
best-fit line created a great opportunity to compare general equations of lines to the special
subset of direct proportions. And their attention
was better since it was “their equation” rather
than one I made up.
What have you done prior to this lesson to prepare
the students to engage in this type of activity?
At the beginning of the year I read through the
directions for an activity with them and discussed in
detail the expectations for the activity. I also reminded
them it is much like doing a lab in science. So by now
my students have a lot of experience working in
groups, reading directions for an exploration, and
being expected to explain their reasoning. We also
spent time in other modules doing a lot of graphing,
by hand and using technology. Areas and volumes,
and converting metric units, have also been previously
studied.
What are some other examples of how you collaborate with the science teacher?
The biggest collaboration the science teacher and I
make has to do with using the same vocabulary—or
relating the vocabulary commonly used in the math
room with the vocabulary commonly used in the science room. We also use some of the same contexts. In
biology, she uses Punnett squares to study genetics. In
math, I start with Punnett squares and expand to using
tree diagrams to study the probability inherent in
genetics. In physics, vectors are used to describe forces
and speed. In math, when studying triangle trigonometry, I use forces and directional speeds as applications. And also in physics, a lot of formulas are needed.
I use physics formulas when practicing solving an
equation for a particular variable, and when working
with algebraic substitution.
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Workshop 7
Notes
Workshop 7
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Insights Into Algebra 1